Intrinsic Axion Statistical Topological Insulator

Ensembles that respect symmetries on average exhibit richer topological states than those in pure states with exact symmetries, leading to the concept of intrinsic average symmetry-protected topological states (ASPTs). The free-fermion counterpart of ASPT is the so-called statistical topological insulator (STI) in disordered ensembles. In this work, we demonstrate the existence of intrinsic STI characterized by the half-quantized magneto-electric polarization P3 = θ/(2π). A C4T symmetry reverses the sign of θ angle, hence seems to protect a Z2 classification of θ =0,π. However, we prove that, if (C4T)^4 =1, the topological state with θ =π is forbidden in band insulators where C4T is exact. Surprisingly, using a real space construction (topological crystal), we find that an axion STI with θ =π can arise in Anderson insulators with disorders respecting C4T on average. To illustrate this state, we construct a lattice model and examine its phase diagram up to the largest numerically accessible system size. An STI phase is identified through gapless surface states and a half-quantized magneto-electric polarization in the bulk. As expected, an unavoidable gapless phase separates the STI from both clean insulators and trivial Anderson insulators, revealing the intrinsicality of the STI. Moreover, we argue that the intrinsicality is robust against electron-electron interactions, i.e., interactions cannot open an adiabatic path connecting the STI to a gapped clean system. Thus, our work provides the first intrinsic crystalline ASPT and its lattice realization. We also generalize the discussion to other crystalline symmetries.